In my experience, the biggest problem for something like a Honda Fit would be ground clearance. At <6" it can have trouble with some potholes, never mind if you need to go over a curb or go off-road at all. It also lacks any sort of physical limited slip, locking differential, 4x4, and/or awd system.....again severely limiting where it can go.

The second biggest issue is the size of the fuel tank. At 10.6 gallons, you're lucky to get 350 miles to a tank, even with the Fits relatively good fuel economy. (Which you won't get if you have to load the vehicle up and travel in traffic to get out of a dangerous area.) Your average pickup or suv gets worse fuel economy, but has a significantly greater driving range (thus giving you a better chance of getting far away from a dangerous area before having to worry about refueling).

The final hurdle is durability. The Fit simply wasn't designed for very rigorous use. Payload capacity is only about 800lbs depending on equipment level. You throw a couple people in there with gear and survival equipment and you can easily max that out. That, in turn, puts a lot of stress on some very important parts of the vehicle (like the powertrain, frame, and crummy low-rolling resistance tires they use). It's also going to reduce the ground clearance further, making the vehicle even more vulnerable. By comparison, something like even Ford's little Transit Connect commercial van is rated for double that amount of payload, as it was designed with more rigorous usage in mind.

All in all, unless you are going to do extensive modifications to the Fit, I would say there are plenty of better choices out there.

A better choice from Honda would be their CR-V. It has more interior room, greater range, greater payload, more ground clearance, it can tow up to 1500lbs, has AWD, has a better weight distribution, and is still plenty small and nimble. On the downside it gets slightly worse fuel economy (25 combined vs. 30 combined) and is more expensive, but I feel both are worthy trade-offs for what one gets in return.